Applications of reproducing kernel theory -development and implementation for engineering and medical problems-

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05357
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Gunma University
• Principal Investigator: Matsuura Tsutomu 群馬大学, 数理データ科学教育研究センター, 教授 (80181692)
• Co-Investigator(Kenkyū-buntansha): 齋藤 三郎 群馬大学, その他部局等, 名誉教授 (10110397)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 再生核理論 / 逆問題 / 画像解析 / 計算アルゴリズム / 医用画像 / ニューラルネットワーク / 機械学習 / ゼロ除算

Outline of Final Research Achievements

We developed a new type of Tikhonov regularization combined with the theory of reproducing kernels. Applying this method to some inverse problems in engineering and medical problems, we constructed calculation algorithms for those problems and confirmed its usefulness by numerical experiments.
These results in the engineering and medical field were presented at 5 academic conferences and published as 2 papers. At the Society of Mathematics, we made 35 presentations including 12 international conferences and 8 invited talks. And we published our mathematical results as 18 papers.

Free Research Field

再生核理論

Academic Significance and Societal Importance of the Research Achievements

物理や工学，医学などの広範囲の分野には，現実問題を扱う際に非常に重要ではあるが，（たとえ解の存在や一意性が証明されていたとしても）具体的に解くことは非現実的で困難だと考えられてきた幾多の逆問題群がある．我々の方法を用いて，これらの問題に対する実際的で有効なアルゴリズムを確立し，これらの逆問題群を高精度・高効率で解くプログラムを現場の計算機に実装できれば，斯界への学問的な貢献のみならず，医学，工学分野や更に産業界への波及効果は非常に大きいものとなる．